Graph Polynomials and Graph Transformations in ... - ELTE TTK TEO

Kelmans studied the following problem when he introduced his transformation. LetR k q(G) be the probability that if we remove the edges of the graph G with probability q,independently of each other, then the obtained random graph has at most k components.Kelmans showed that the Kelmans transformation decreases this probability for every q, inother words, R k q(G ′ ) ≤ R k q(G). Satyanarayana, Schoppmann and Suffel [10] rediscoveredthis result and they proved that the Kelmans transformation decreases the number ofspanning trees: τ(G ′ ) ≤ τ(G). We proved the following results.Theorem 3.2. [1] Let G ′ be obtained from G by a Kelmans transformation. Let µ(G)and µ(G ′ ) denote the largest eigenvalue of the adjacency matrix of G and G ′ , respectively.Then µ(G ′ ) ≥ µ(G).This result enabled us to attain a breakthrough in an old problem of Eva Nosal. In thisproblem one has to bound the expression µ(G) + µ(G) in terms of the number of vertices.We managed to prove the following theorem which was a significant improvement of theprevious results.Theorem 3.3. [1] Let G be a graph on n vertices. Thenµ(G) + µ(G) ≤ 1 + √ 3n.2We managed to prove the following theorems concerning graph polynomials.Theorem 3.4. [5] Let M(G,x) be the matching polynomial of the graph G:M(G,x) = ∑ r=0(−1) r m r (G)x n−2r .3Let t(G) denote the largest root of the matching polynomial. Let G ′ be obtained from G bya Kelmans transformation.Then m k (G ′ ) ≤ m k (G) holds for every 1 ≤ k ≤ n/2 and t(G ′ ) ≥ t(G).Theorem 3.5. [5] Let I(G,x) be the independence polynomial of the graph G:I(G,x) = ∑ k=0(−1) k i k (G)x k .Let β(G) denote the smallest root of the independence polynomial. Let G ′ be obtained fromG by a Kelmans transformation.Then i k (G ′ ) ≥ i k (G) holds for every 1 ≤ k ≤ n and β(G ′ ) ≤ β(G).Theorem 3.6. [5] Let L(G,x) = ∑ nk=1 (−1)n−k a k (G)x k be the Laplacian polynomial ofthe graph G. Let G ′ be obtained from G by a Kelmans transformation.Then a k (G ′ ) ≤ a k (G) for 1 ≤ k ≤ n.Theorem 3.7. [5] Let ch(G,x) = ∑ nk=1 (−1)n−k c k (G)x k be the chromatic polynomial ofthe graph G. Let G ′ be obtained from G by a Kelmans transformation.Then c k (G ′ ) ≤ c k (G) for 1 ≤ k ≤ n.4. Generalized tree shiftWe define the generalized tree shift as follows.